Suppose that xandy vary inversely. Write a function that models each inverse variation. Graph the function and fi nd ywhenx5 10.

(1)

8-1

Additional Vocabulary Support

_{Inverse Variation}

Choose the expression from the list that best matches each sentence.

combined constant of inverse joint variation variation variation variation

1. equations of the form xy 5k

2. when one quantity varies with respect to two or more quantities

3. when one quantity varies directly with two or more quantities

4. the product of two variables in an inverse variation

Choose the expression from the list that best matches each sentence.

combined constant of inverse joint variation variation variation variation

5. When one quantity increases and the other quantity decreases proportionally, the relationship is an .

6. Th e function z 5kxyis an example of a .

7. Th e is represented by the variable k.

8. Both functions z 5_wykx and z5 kxyare examples of .

Multiple Choice

9. Which function would be used to model the relationship “x and y vary inversely”?

y 5kx z 5kxy y 5k_x y5 x

k

10. Which function would be used to model the relationship “z varies jointly with

x and y”?

y 5kxz z 5kxy y 5_xzk y5 xz

k

inverse variation

joint variation

constant of variation

combined variation inverse variation

combined variation

joint variation

constant of variation

C

(2)

equation of the form PV5k. Estimate P when V5 62.

Understanding the Problem

1. Th e data can be modeled by

z

.

2. What is the problem asking you to determine?

Planning the Solution

3. What does it mean that the data can be modeled by an inverse variation?

.

4. How can you estimate the constant of the inverse variation?

.

5. What is the constant of the inverse variation?

6. Write an equation that you can use to fi nd P when V562.

Getting an Answer

7. Solve your equation.

8. What is an estimate for P when V562?

8-1

Think About a Plan

_{Inverse Variation}

A B

140.00 100 2

3 147.30 95 4 155.60 90

164.70 85 5

175.00 80 6

186.70 75 7

1 P V

PV5k

an estimate of the value of P when V562

about 14,000

about 226

62P514,000 The product of each pair of P and V values is approximately the same constant

Find PV for each row of the data. It should be approximately the same for

each row

62P514,000

(3)

8-1

Practice

Form G

Inverse Variation

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.

1. x

y 2 10

4 5

5 4

20 1

2. x

y 1 2

3 8

7 20

10 29

3. x

y 1

6 2

12 5

30 7

42

4. x

y 0.2

25 0.5

62.5 2

250 3

375

5. x

y 31 7 3 2

1 10

1 2

3 2

5 2

6. x

y 3 5

1.5 10

0.5 30

0.3 50

Suppose that x and y vary inversely. Write a function that models each inverse variation. Graph the function and fi nd y when x510.

7. x57 when y 52 8. x54 when y 50.2 9. x51₃ when y5 ₁₀9

10. Th e students in a school club decide to raise money by selling hats with the school mascot on them. Th e table below shows how many hats they can expect to sell based on how much they charge per hat in dollars.

Price per Hat (p) 5 6 8 9

72 60 45 40 Hats Sold (h)

a. What is a function that models the data?

b. How many hats can the students expect to sell if they charge $7.50 per hat?

11. Th e minimum number of carpet rolls n needed to carpet a house varies directly as the house’s square footage h and inversely with the square footage

r in one roll. It takes a minimum of two 1200-ft2 carpet rolls to cover 2300 ft2 of fl oor. What is the minimum number of 1200-ft2 carpet rolls you would need to cover 2500 ft2 of fl oor? Round your answer up to the nearest half roll.

y514_x; 7₅ y5_5x4; 0.08 or ₂₅2 y5_10x3 ; 0.03 or ₁₀₀3

ph5360 or h5360_p

48

2.5 inverse; y520_x neither

direct; y5125x

inverse; y515_x direct; y56x

(4)

8-1

Practice

(continued) Form G

Inverse Variation

12. On Earth, the mass m of an object varies directly with the object’s potential energy E and inversely with its height above the Earth’s surface h. What is an equation for the mass of an object on Earth? (Hint:E5gmh, where g is the acceleration due to gravity.)

Each ordered pair is from an inverse variation. Find the constant of variation.

13. Q3, 1₃R 14. (0.2, 6) 15. (10, 5) 16. Q5₇, 2₅R

17. (213, 22) 18. Q1₂, 10R 19. Q1₃, 6₇R

20. (4.8, 2.9) 21. _Q5₈,22₅_R 22. (4.75, 4)

Write the function that models each variation. Find zwhen x5 6 and y5 4.

23. z varies jointly with x and y. When x5 7 and y5 2, z5 28.

24. z varies directly with x and inversely with the cube of y. When x5 8 and y5 2,

z5 3.

Each pair of values is from an inverse variation. Find the missing value.

25. (2, 4), (6, y) 26. Q1₃, 6R, Qx, 21₂R 27. (1.2, 4.5), (2.7, y)

28. One load of gravel contains 240 ft3 of gravel. Th e area A that the gravel will cover is inversely proportional to the depth d to which the gravel is spread. a. Write a model for the relationship between the area and depth for one load

of gravel.

b. A designer plans a playground with gravel 6 in. deep over the entire play area. If the play area is a rectangle 40 ft wide and 24 ft long, how many loads of gravel will be needed?

m5kE_h, where k51_g

2286

1 1.2 50 2₇

13.92

5

21₄

2 7

19

24 2

2 loads 4

3

z 5 2xy; 48

z5 3x y3;

9 32

(5)

8-1

Practice

Form K

Inverse Variation

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write an equation to model the direct and inverse variations.

1. _x _y

0.1 3 6 24

3 0.1 0.05 0.0125

2. _x _y

1 2 5 6

3 6 15 18

3. _x _y

0 2 4 6

1 5 7 8

Suppose that x and y vary inversely. Write a function that models each inverse variation. Graph the function and fi nd y when x 510.

4. x52 when y 5 24 5. x5 29 when y 5 21 6. x51.5 when y 510

7. Suppose the table at the right shows the time t it takes to drive home when you travel at various average speeds s.

a. Write a function that models the relationship between the speed and the time it takes to drive home.

b. At what speed would you need to drive to get home in 50 min or 5₆ h?

Time t (h) Speed s (mi/h) 60

40

30

13.3

1 6 1 4 1 3 3 4

O

x y 4

4 8

4 O x

y

4

4 4 O x

y

4

4 4 4 inverse variation; y50.3_x

y5 28_x; 24₅

direct variation; y53x

y59_x; ₁₀9

s510_t

12 mi/h

neither

(6)

8. Th e height h of a cylinder varies directly with the volume of the cylinder and inversely with the square of the cylinder’s radius r with the constant equal to 1_p. a. Write a formula that models this combined variation.

b. What is the height of a cylinder with radius 4 m and volume 500 m3? Use 3.14 for p and round to the nearest tenth of a meter.

9. Some students volunteered to clean up a highway near their school. Th e amount of time it will take varies directly with the length of the section of highway and inversely with the number of students who will help. If 25 students clean up 5 mi of highway, the project will take 2 h. How long would it take 85 students to clean up 34 mi of highway?

Write the function that models each variation. Find z when x52 and y56.

10. z varies inversely with x and directly with y. When x 55 and y5 10, z5 2.

11. z varies directly with the square of x and inversely with y. When x52 and

y5 4, z 53.

Each ordered pair is from an inverse variation. Find the constant of variation.

12. (2, 2) 13. (1, 8) 14. (9, 4)

Each pair of values is from an inverse variation. Find the missing value.

15. (9, 5), (x, 3) 16. (8, 7), (5, y) 17. (2, 7), (x, 1)

8-1

Practice

(continued) Form K

Inverse Variation

h5 V

πr2

z5y_x; 3

z53x_y2; 2

k54

x515

k58

y511.2

k536

x514 10.0 m

(7)

8-1

Standardized Test Prep

_{Inverse Variation}

Multiple Choice

For Exercises 1−5, choose the correct letter.

1. Which equation represents inverse variation between x and y?

4y5 kx xy 54k y 54kx 4k 5x_y

2. Th e ordered pair (3.5, 1.2) is from an inverse variation. What is the constant of variation?

2.3 2.9 4.2 4.7

3. Suppose x and y vary inversely, and x5 4when y 59. Which function models the inverse variation?

y 536_x x 5₃₆y y 5₃₆x x_y 536

4. Suppose x and y vary inversely, and x5 23 wheny 51₃. What is the value of

y when x 59?

29 21 21₉ 1₉

5. In which function does t vary jointly with q and r and inversely with s?

t 5kq_rs t 5_qrks t 5 s

kqr t5

kqr s

Short Response

6. A student suggests that the graph at the right represents the inverse variation y 53_x. Is the student correct? Explain.

2 2 4 6

4 6 8

y

x

B

H

A

H

D

No; for each point on the graph xy56, not 3. [2] correct answer with explanation

(8)

8-1

Enrichment

_{Inverse Variation}

Each situation below can be modeled by a direct variation, inverse variation, joint variation, or combined variation equation. Decide which model to use and explain why.

1. Th e circumference C of a circle is about 3.14 times the diameter d.

2. Th e number of cavities that develop in a patient’s teeth depends on the total number of minutes spent brushing.

3. Th e time it takes to build a bridge depends on the number of workers.

4. Th e number of minutes it will take to solve a problem set depends on the number of problems and the number of people working on the problem set.

5. Th e current I in an electrical circuit decreases as the resistance R increases.

6. Charles’s Gas Law states the volume V of an enclosed gas at a constant pressure will increase as the absolute temperature T increases.

7. Boyle’s Law states that the volume V of an enclosed gas at a constant temperature is related to the pressure. Th e pressure of 3.45 L of neon gas is 0.926 atmosphere (atm). At the same temperature, the pressure of 2.2 L of neon gas is 1.452 atm.

Direct variation; answers may vary. Sample: This relationship can be modeled by the equation C 5 3.14d, where 3.14 is the constant of variation.

Inverse variation; answers may vary. Sample: As the number of minutes spent brushing increases, the number of cavities should decrease. This suggests an inverse variation.

Inverse variation; answers may vary. Sample: As the number of workers increases, the time it takes to build a bridge should decrease. This suggests an inverse variation.

Combined variation; answers may vary. Sample: The time to solve a problem set increases as the number of problems increases, but decreases as the number of people working on the set increases. This suggests a combined variation.

Inverse variation; answers may vary. Sample: As one variable increases, the other decreases. This suggests an inverse variation.

Direct variation; answers may vary. Sample: As one variable increases, so does the other. This suggests a direct variation.

(9)

8-1

Reteaching

_{Inverse Variation}

Th e fl owchart below shows how to decide whether a relationship between two variables is a direct variation, inverse variation, or neither.

How does the value of x change?

How does the value of y change? How does the value of y change?

yes _yes

no no

increases decreases

increases decreases increases

neither direct variation inverse variation neither decreases

Test x and y values in direct y x variation model: _k_.

Test _x and _y values in inverse variation model: xy k.

Is each ratio of _y to _x equal to the same value, _k?

Is each product of _x and _y equal to the same value, _k?

Problem

Do the data in the table represent a direct variation, inverse variation, or neither?

x y

1

20 2

10 4

5 5

4

As the value of x increases, the value of y decreases, so test the table values in the inverse variation model: xy 5k : 1 ? 20520, 2? 10520, 4 ? 55 20, 5 ? 4 520. Each product equals the same value, 20, so the data in the table model an inverse variation.

Exercises

Do the data in the table represent a direct variation, inverse variation, or neither?

1. x

y 5

10 10

20 15

30 20

40 2.

x y

1

12 3

4 4

3 6

2

(10)

8-1

Reteaching

(continued)

Inverse Variation

To solve problems involving inverse variation, you need to solve for the constant of variation k before you can fi nd an answer.

Problem

Th e time t that is necessary to complete a task varies inversely as the number of people p working. If it takes 4 h for 12 people to paint the exterior of a house, how long does it take for 3 people to do the same job?

t 5_pk Write an inverse variation. Because time is dependent on people, t is the dependent variable and p is the independent variable.

4 5₁₂k Substitute 4 for t and 12 for p.

485k Multiply both sides by 12 to solve for k, the constant of variation.

t 548_p Substitute 48 for k. This is the equation of the inverse variation.

t 548₃ 516 Substitute 3 for p. Simplify to solve the equation.

It takes 3 people 16 h to paint the exterior of the house.

Exercises

3. Th e time t needed to complete a task varies inversely as the number of people

p. It takes 5 h for seven men to install a new roof. How long does it take ten men to complete the job?

4. Th e time t needed to drive a certain distance varies inversely as the speed r. It takes 7.5 h at 40 mi/h to drive a certain distance. How long does it take to drive the same distance at 60 mi/h?

5. Th e cost of each item bought is inversely proportional to the number of items when spending a fi xed amount. When 42 items are bought, each costs $1.46. Find the number of items when each costs $2.16.

6. Th e length l of a rectangle of a certain area varies inversely as the width w. Th e length of a rectangle is 9 cm when the width is 6 cm. Determine the length if the width is 8 cm.

3.5 h

5 h

about 28 items

(11)

8-2

Additional Vocabulary Support

_{The Reciprocal Function Family}

For Exercises 1–5, draw a line from each word in Column A to its defi nition in Column B.

Column A Column B

1. reciprocal A. function that models inverse variation

2. branch B. stretches, compressions, refl ections, and horizontal and vertical translations

3. reciprocal function C. multiplicative inverse

4. refl ection of the reciprocal function D. the graph of y 5 21_x

5. transformations E. each part of the graph of a reciprocal function

For Exercises 6–9, the graph of each function is a transformation of the parent

graph of f(x)51_x. Draw a line from each function to its transformation.

6. f(x) 52_x A. a horizontal translation

7. f(x) 5 21_x B. a refl ection over the x-axis 8. f(x) 5_x ₂1 ₂ C. a vertical translation

(12)

Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. a. Gasoline Mileage Suppose you drive an average of 10,000 miles each year.

Your gasoline mileage (mi/gal) varies inversely with the number of gallons of gasoline you use each year. Write and graph a model for your average mileage

m in terms of the gallons g of gasoline used.

b. After you begin driving on the highway more often, you use 50 gal less per year. Write and graph a new model to include this information.

c. Calculate your old and new mileage assuming that you originally used 400 gal of gasoline per year.

1. Write a formula for gasoline mileage in words.

.

2. Write and graph an equation to model your average mileage m

in terms of the gallons g of gasoline used.

3. Write and graph an equation to model your average mileage m in terms of the gallons g of gasoline used if you use 50 gal less per year.

4. How can you fi nd your old and your new mileage from your equations?

.

5. What is your old mileage?

6. What is your new mileage?

8-2

Think About a Plan

_{The Reciprocal Function Family}

0 100 200 300 400 0

10 20 30 40

Gas Used (gal)

Mileage (mi/gal)

m

g

0 100 200 300 400 0

10 20 30 40

m

g

Gas Used (gal)

Mileage (mi/gal)

The mileage is equal to the number of miles divided by the number of gallons

Evaluate each equation at g5400

25 mi/gal

about 28.6 mi/gal m510,000_g

(13)

8-2

Practice

Form G

The Reciprocal Function Family

Graph each function. Identify the x- and y-intercepts and the asymptotes of the graph. Also, state the domain and the range of the function.

1. y5 12_x 2. y5 5_x 3. y5 24_x

Use a graphing calculator to graph the equations y51_x and y5a_x using the given value of a. Th en identify the eff ect of a on the graph.

4. a53 5. a5 25 6. a50.4

Sketch the asymptotes and the graph of each function. Identify the domain and range.

7. y5 1_x 13 8. y5 ₄3_x 11₂ 9. y5 _x₂3 ₁ 12

Write an equation for the translation of y5 23_x that has the given asymptotes. 10. x5 21; y 53 11. x54; y 5 22 12. x50; y 5 6

6 3 6 9 9

3 6 9

x y

O

x scale: 1_y scale: 1 _x scale: 1_y scale: 1

4 4

x y

O

x scale: 0.1y scale: 0.1 no intercepts; x50,

y50; all real numbers except x50; all real num. except y50

Stretch by a factor of 3.

all real numbers except x50; all real numbers except y53

all real numbers except x50; all real numbers except y51₂

all real numbers except x51; all real numbers except y52

no intercepts; x50, y50; all real numbers except x50; all real num. except y50

Reﬂ ect over x-axis and stretch by a factor of 5.

no intercepts; x50, y50; all real numbers except x50; all real num. except y50

Shrink by a factor of 0.4.

y5 2_x₁3 ₁1 3 y5 2_x₂3 ₄22 y5 23_x16 x

y

O

1 2 2

2 3 4

x y

O 4 4

4 4 6

x y

O 1 2 1 2

2

3

O 2

2 2

(14)

8-2

Practice

(continued) Form G

The Reciprocal Function Family

13. Th e length of a pipe in a panpipe / (in feet) is inversely proportional to its pitch p (in hertz). Th e inverse variation is modeled by the equation p 5495_/ . Find

the length required to produce a pitch of 220 Hz.

Write each equation in the form y5k_x.

14. y5 ₅4_x 15. y5 2₂7_x 16. xy 5 20.03

Sketch the graph of each function.

17. xy 56 18. xy 11050 19. 4xy 5 21

20. Th e junior class is buying keepsakes for Class Night. Th e price of each

keepsake p is inversely proportional to the number of keepsakes s bought. Th e keepsake company also off ers 10 free keepsakes in addition to the class’s order. Th e equation p 5_s1800₁₁₀ models this inverse variation.

a. If the class buys 240 keepsakes, what is the price for each one?

b. If the class pays $5.55 for each keepsake, how many can they get, including the free keepsakes?

c. If the class buys 400 keepsakes, what is the price for each one? d. If the class buys 50 keepsakes, what is the price for each one?

Graph each pair of functions. Find the approximate point(s) of intersection.

21. y5 _x₂3 ₄; y 52 22. y5 _x₁2 ₅; y 5 21.5 2 6

O 2 6

x y

2 O

2 4

4 8

x y

0.5 O 0.5 1

1

x y 2.25 ft

$7.20

324

$4.39

(5.5, 2) (26.3, 21.5) $30

y50.8_x y523.5_x y520.03_x

Intersection x5 5.5 y5 2 x scale: 1 y scale: 1

Intersection

(15)

8-2

Practice

Form K

The Reciprocal Function Family

Graph each function. Identify the x- and y-intercepts and asymptotes of the graph. Also, state the domain and range of the function.

1. y5 22_x 2. y5 4_x 3. y5 25_x

Graph the equations y51_x and y5a_x using the given value of a. Th en identify the eff ect of a on the graph.

4. a5 23 5. a54 6. a5 20.25

Sketch the asymptotes and the graph of each function. Identify the domain and range.

7. y5 1_x 12 8. y5 _x₂1 ₂ 13 9. y5 _x2₁10₁ 28

O x y 4 8 4 8 4 8

4 8 O x

y 4 4 8 8 4 O x y 8 4 8 4 8 8 x-intercept: none;

y-intercept: none; horizontal asymptote: x-axis; vertical asymptote: y-axis; domain: all real numbers except x50; range: all real numbers except y50

reﬂ ected across the x-axis and stretched by a factor of 3

domain: all real numbers except 0; range: all real numbers except 2

x-intercept: none;

y-intercept: none; horizontal asymptote: x-axis; vertical asymptote: y-axis; domain: all real numbers except

x50; range: all real numbers except y50

stretched by a factor of 4

domain: all real numbers except 2; range: all real numbers except 3

x-intercept: none;

y-intercept: none; horizontal asymptote: x-axis; vertical asymptote: y-axis; domain: all real numbers except x50; range: all real numbers except y50

reﬂ ected across the x-axis and compressed by a factor of 0.25

domain: all real numbers except 21; range: all real numbers except 28

O x y 4 8 4 8 4 8 4 8

yⴝ1_x yⴝⴚ3_x

O x y 4 4 8 4 8 4

yⴝ1_x yⴝ4_x

O x y 4 8 4 8 4 8 4 8

yⴝ1_x yⴝⴚ0.25_x

O x y 4 4 8 4 8

4 O x

(16)

Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. Write an equation for the translation of y53_x that has the given asymptotes.

10. x50 and y 52 11. x5 22 and y 54 12. x55 and y 5 23

Sketch the graph of each function.

13. 3xy 51 14. xy 28 50 15. 2xy 5 26

16. Writing Explain the diff erence between what happens to the graph of the parent function of y 5a_x when uau .1 and what happens when 0 , uau ,1.

17. Suppose your class wants to get your teacher an end-of-year gift of a weekend package at her favorite spa. Th e package costs $250. Let c equal the cost each student needs to pay and s equal the number of students.

a. If there are 22 students, how much will each student need to pay? b. Using the information, how many total students (including those from

other classes) need to contribute to the teacher’s gift, if no student wants to pay more than $7?

c. Reasoning Did you need to round your answers up or down? Explain.

8-2

Practice

(continued) Form K

The Reciprocal Function Family

y53_x 12

When »a… S1, the parent function is stretched by the factor of a. When 0 R »a… R1, the parent function is compressed by the factor of a.

up; because if you round down, the total contribution by the students would be less than $250.

y5_x₁3 ₂14 y5_x₂3 ₅23

O

x y 4 8

4 8

4 8 O x

y

4

4 4 O

x y

4 8

$11.37

(17)

8-2

Standardized Test Prep

_{The Reciprocal Function Family}

Multiple Choice

1. What is an equation for the translation of y 5 24.5_x that has asymptotes at

x53 and y 5 25?

y 5 2_x4.5₂₃25 y 5 2_x4.5₂₅13

y 5 2_x4.5₁₃25 y 5 2_x4.5₁₅13

2. What is the equation of the vertical asymptote of y 5 _x₂2 ₅?

x 5 25 x 50 x5 2 x55

3. Which is the graph of y 5_x ₁1 ₁2 2?

Extended Response

4. A race pilot’s average rate of speed over a 720-mi course is inversely proportional to the time in minutes t the pilot takes to fl y a complete race course. Th e pilot’s fi nal score s is the average speed minus any penalty points

p earned.

a. Write a function to model the pilot’s score for a given t and p. (Hint:d5 rt) b. Graph the function for a pilot who has 2 penalty points.

c. What is the maximum time a pilot with 2 penalty points can take to fi nish the course and still earn a score of at least 3?

x y O

1 2 1

1

2

4

3

5

2 2

4

2

4

x y

O

2

3 1 1

3

4

5

x y O O

2

4

4 1 y _x

A

I

D

[4] s5720_t 2p; [3] correct answer with most of work shown and appropriate strategies used OR incorrect answer with all work shown and appropriate strategies used

[2] correct answer with little work shown OR incorrect answer but work shown reﬂ ects some understanding of problem

[1] answer is incomplete or incorrect and no work is shown

[0] no answer given 144 min

100 200 300 2

4 s

O

(18)

8-2

Enrichment

_{The Reciprocal Function Family}

Understanding Horizontal Asymptotes

Th e line y 5 3₄ is a horizontal asymptote for the graph of the

function y 53₄x_x1₂5₈. By using long division, you can rewrite

this function in the form quotient 1 remainder divided by

the divisor: y 5 3₄1₄_x11₂₈.

Examine what happens to the remainder divided by the divisor and the value of y as the value of x gets larger. Fill in the following table to four decimal places.

Note that as x gets larger, both the remainder and the value of y get smaller.

Although the value of y is always greater than 3₄, it gets closer to 3₄ as x gets

larger. As x gets infi nitely large, y approaches 3₄ from above. Write this as:

As x S 1`, yS3₄ from above.

Examine what happens as x gets smaller. Fill in the following table to four decimal places.

Here the value of y is always less than 3₄, but it gets closer to 3₄ as x gets smaller (more negative). Write this as: As x S2`, yS3₄ from below.

In both cases, y approaches 3₄, so the horizontal asymptote is y 53₄.

O

4 6

2

4

4 4 x

y

1. 2. 3.

x

3 10 100

11 4x 2 8

3 4

11 4x 2 8

y5 1

4. 5. 6.

x

23

210

2100

11 4x 2 8

3 4

11 4x 2 8

y5 1

2.7500 3.5000

0.3438 1.0938

0.0281 0.7781

20.5500 0.2000

2_0.2292 _0.5208

(19)

8-2

Reteaching

_{The Reciprocal Function Family}

A Reciprocal Function in General Form

Two Members of the Reciprocal Function Family

Th egeneral form is y 5 a

x 2h 1k, where

a20 and x2h.

When a21, h 50, and k 50, you get the inverse variation function,y 5a_x.

Th e graph of this equation has a horizontal asymptote at y 5k and a vertical asymptote at x5 h.

When a 51, h 50, and k 50, you get the parent reciprocal function,y 51_x.

Problem

What is the graph of the inverse variation function y 52_x5?

Step 1 Rewrite in general form and identify a, h, and k.

y 5_x2₂5₀10 a5 25, h50, k5 0

Step 2 Identify and graph the horizontal horizontal asymptote: y 5k

and vertical asymptotes. y 50

vertical asymptote: x 5h

x 50

Step 3 Make a table of values for y 52_x5. Plot the points and then connect the points in each quadrant to make a curve.

y

x 25 22.5 21 1 5

1 2 21 2.5

22

5 25

Exercises

Graph each function. Include the asymptotes.

1. y5 9_x 2. y5 24_x 3. xy 52

5 4

y

4

4 3

3 3 2

2 2 1

1

5 3 1

x

x y

O

4

6 2 6

2 6

6 4

9

3

9

3 6

6 9

9 y

O

x

2

1

2

1 2

2

y

(20)

2h 1 k is a translation of the inverse

variation function y5 a_x. Th e translation is h units horizontally and k units vertically. Th e translated graph has asymptotes at x5 h and y 5k.

Problem

What is the graph of the reciprocal function y5 2_x₁6 ₃ 12?

Step 1 Rewrite in general form and identify a, h, and k.

y 5 26

x 2(23)1 2 a5 26, h 5 23, k 52

Step 2 Identify and graph the horizontal horizontal asymptote: y5 k

and vertical asymptotes. y5 2

vertical asymptote: x5 h

x5 23

Step 3 Make a table of values for y 52_x6, then translate each (x, y) pair to (x1h, y 1k). Plot the translated points and connect the points in each quadrant to make a curve.

x 1₍2₃₎

y 12 y

x 26

29

1

3

6

3

21

1

23

26

2

4

3

0

22

0

22

25

3

5 2

21

23

21

Exercises

4. y5 _x₂3 ₂ 24 5. y5 2_x₂4 ₈ 6. y5 ₃2_x 13₂ 8

8 y

4 6 6

6 4

4 2

8 6 2

x

8-2

Reteaching

(continued)

The Reciprocal Function Family

O 4

6 8 10

4 8x

y

O y

2 4 6

4

x

2 4 6 8

2 4

2 2

2 x y

(21)

8-3

Additional Vocabulary Support

_{Rational Functions and Their Graphs}

Concept List

Choose the concept from the list above that best represents the item in each box.

1. the line that a graph approaches as y

increases in absolute value

2. In the denominator, these reveal the points of discontinuity.

3. Th is type of

discontinuity appears as a hole in the graph.

4. Th is type of graph has no jumps, breaks, or holes.

5. a function that you can write in the form

f(x)5 P(x)

Q(x) where P(x) and Q(x) are

polynomial functions

6. a graph that has a one-point hole or a vertical asymptote

7. Th is type of

discontinuity appears as a vertical asymptote on the graph.

8. Th e graph of f (x) is not continuous at this point.

9. the line that a graph approaches as x

increases in absolute value

continuous discontinuous factors

horizontal asymptote non-removable discontinuity point of discontinuity rational function removable discontinuity vertical asymptote

vertical asymptote

continuous

non-removable discontinuity

factors

rational function

point of discontinuity

removable discontinuity

discontinuous

(22)

Grades A student earns an 82% on her fi rst test. How many consecutive 100% test scores does she need to bring her average up to 95%? Assume that each test has equal impact on the average grade.

Understanding the Problem

1. One test score is

z

.

2. Th e average of all the test scores is

z

.

3. What is the problem asking you to determine?

Planning the Solution

4. Let x be the number of 100% test scores. Write an expression for the total number of test scores.

5. Write an expression for the sum of the test scores.

6. How can you model the student’s average as a rational function?

Getting an Answer

7. How can a graph help you answer this question?

.

8. What does a fractional answer tell you? Explain.

.

9. How many consecutive 100% test scores does the student need to bring her average up to 95%?

8-3

Think About a Plan

_{Rational Functions and Their Graphs}

82%

95%

the number of tests with scores of 100% the student needs to have an

average of 95%

Answers may vary. Sample: I can graph the rational function and the

function y 5 95 at the same time. The intersection is the solution

I would have to round a fractional answer up to the nearest whole number, because the

number of test scores must be a whole number and the average must be 95% or greater x1 1

3 100x1 82

(23)

8-3

Practice

Form G

Rational Functions and Their Graphs

Find the domain, points of discontinuity, and x- and y-intercepts of each rational function. Determine whether the discontinuities are removable or nonremovable.

1. y5 (x 2_x4)(₁x₃13) 2. y5 (x 2_x3)(₂x₂11)

3. y5 _x₁2 ₁ 4. y5 4x

x4116

Find the vertical asymptotes and holes for the graph of each rational function.

5. y5 52x

x221 6. y5

x222

x12

7. y5 x

x(x 21) 8. y5

x13

x229

9. y5 x 22

(x 12)(x22) 10. y5

x224

x214

11. y5 x_x22₂25₄ 12. y5 (x 22)(2x 13) (5x 14)(x 23)

Find the horizontal asymptote of the graph of each rational function.

13. y5 _x₂2 ₆ 14. y 5x_x ₂12₄ 15. y5 2x2 13

x2 26 16. y 5

3x212

x222

Sketch the graph of each rational function.

17. y5 _x₂3 ₂ 18. y 5 3

(x22)(x 12) 19. y5

x

x214 20. y 5

x 12

x 21 all real num except x 5 23;

(4, 0), (0, 24); removable

all real num except 21; x 5 21, no x-intercept, (0, 2); non-removable

vertical asymptotes at x51 and x5 21

vertical asymptote at x51; hole at x50

vertical asymptote at x5 22; hole at x52

vertical asymptote at x54

y50 y51

all real num except 2; x 5 2; (21, 0) (3, 0), (0, 3₂); non-removable

all real num; none; (0, 0)

vertical asymptote at x5 22

vertical asymptote at x53; hole at x5 23

no vertical asymptotes or holes

vertical asymptotes at x5 24₅ and x53

y52 y50

x y

O 2

⫺2

⫺4 4 6 8

⫺4 ⫺6 4 6

x y

O 2 ⫺2

2

⫺2 ⫺4

⫺6 4 6

⫺4 ⫺6 4 6

O 2

⫺2 ⫺2

2 x y

O

2

(24)

8-3

Practice

(continued) Form G

Rational Functions and Their Graphs

21. How many milliliters of 0.75% sugar solution must be added to 100 mL of 1.5% sugar solution to form a 1.25% sugar solution?

22. A soccer player has made 3 of his last 24 shots on goal, or 12.5%. How many more consecutive goals does he need to raise his shots-on-goal average to at least 20%?

23. Error Analysis A student listed the asymptotes of the function y 5 x215x 16

x(x2 14x 14) as shown at the right. Explain the student’s error(s). What are the correct asymptotes?

24. y5 x

x(x 26) 25. y 5 2x

x 26 26. y5

x221

x224 27. y 5

2x2110x 112

x2 29

28. You start a business word-processing papers for other students. You spend $3500 on a computer system and offi ce furniture. You fi gure additional costs at $.02 per page.

a. Write a rational function modeling the total average cost per page. Graph the function.

b. What is the total average cost per page if you type 1000 pages? If you type 2000?

c. How many pages must you type to bring your total average cost to less than $1.50 per page?

d. What are the vertical and horizontal asymptotes of the graph of the function?

horizontal asymptote none

vertical asymptote x = 0

50 mL

3

The horizontal asymptote should be y50, because the degree of the numerator is less than the degree of the denominator. The zeros of the denominator are x50 and x5 22, so there should also be a vertical asymptote at x5 22.

x50; y50.02 $3.52; $1.77

at least 2365 pages

x y

O 2 ⫺2

2 ⫺2 ⫺4

⫺6 4

⫺4 ⫺6 4 6

x y

3 ⫺3

⫺6 6 9

⫺6 ⫺9 6

y50.02x1_x 3500, where x 5 number of pages x

y

O 2 ⫺2

2

⫺2 4 6 8 10 ⫺4

⫺6 4 6

x y

O ⫺3

3

⫺3

⫺6 9 12

⫺6

(25)

8-3

Practice

Form K

Rational Functions and Their Graphs

Find the domain, points of discontinuity, and x- and y-intercepts of each rational function. Determine whether the discontinuities are removable or non-removable. To start, factor the numerator and denominator, if possible.

1. y5 x_x1₂5₂ 2. y5 1

x212x11 3. y5

x14

x212x28

Find the vertical asymptotes and holes for the graph of each rational function.

4. y5 x_x1₁6₄ 5. y5 (x 2_x2)(₂x₂21) 6. y5 x 11 (3x 22)(x 23)

Find the horizontal asymptote of the graph of each rational function. To start, identify the degree of the numerator and denominator.

7. y5 x_x1₁1₅ 8. y5 x 12

2x2 24 9. y5

3x3 24 4x 11

x11

x15

ddegree 1 ddegree 1

10. y5 x 12

(x 13)(x24) 11. y5

x 13

(x 21)(x25) 12. y5 2x

3x 21 domain: all real numbers

except x52; non-removable point of discontinuity at x52; x-intercept: x5 25; y-intercept: y5 25₂

vertical asymptote: x5 24

y51

domain: all real numbers except x5 21;

non-removable point of discontinuity at x5 21; x-intercept: none; y-intercept: y51

vertical asymptote: none; hole at x52

y50

domain: all real numbers except x52 and x5 24; non-removable point of discontinuity at x52 and removable point of discontinuity at x5 24; x-intercept: none; y-intercept: y5 21₂

vertical asymptotes: x52₃ and x53

no horizontal asymptote

O

x y

4 8

⫺4 ⫺8

4 8 O x

y

4

⫺4 ⫺4 ⫺8

8

4 O x

y

4

⫺4 ⫺8

(26)

13. Th e CD-ROMs for a computer game can be manufactured for $.25 each. Th e development cost is $124,000. Th e fi rst 100 discs are samples and will not be sold. a. Write a function for the average cost of a disc that is not a sample.

b. What is the average cost if 2000 discs are produced? If 12,800 discs are produced?

c. Reasoning How could you fi nd the number of discs that must be produced to bring the average cost under $8?

d. How many discs must be produced to bring the average cost under $8?

14. Error Analysis For the rational function y5 x222x28

x229 , your friend said that the vertical asymptote is x 51 and the horizontal asymptotes are y 53 and y 5 23. Without doing any calculations, you know this is incorrect. Explain how you know.

15. y5 4x22100

2x2 1x215 16. y5 2x2

5x 11 17. y5

2

x224

18. Multiple Choice What are the points of discontinuity for the graph of

y5 (2x 13)(x 25) (x 15)(2x 21)?

25, 1 23₂, 5 25, 1₂ 5, 21₂

8-3

Practice

(continued) Form K

Rational Functions and Their Graphs

y50.25x_x1₂124,000₁₀₀

c. The intersection of y50.25x_x1₂124,000₁₀₀ and y58 rounded to the next whole number gives the number of discs that must be produced.

A rational function can have only 1 horizontal asymptote. Your friend must have switched the horizontal and vertical asymptote answers.

$65.53; $10.02

16,104 discs

O x

y

4 8

⫺8 ⫺4

O x

y

4 8

⫺4 ⫺8

4 8

O x y

2

⫺2 ⫺4

2

(27)

8-3

Standardized Test Prep

_{Rational Functions and Their Graphs}

Multiple Choice

1. What function has a graph with a removable discontinuity at Q5, 1₉R?

y 5 (x 25)

(x14)(x 25) y 5

4x 21 5x 11

y 5_x ₂4 ₅ y 5₅x_x1₂1₄

2. What is the vertical asymptote of the graph of y 5(x12)(x23)

x(x23) ?

x 5 23 x 5 22 x5 0 x53

3. What best describes the horizontal asymptote(s), if any, of the graph of

y5 x212x28 (x16)2 ?

y 5 26 y 51

y 50 Th e graph has no horizontal asymptote.

4. Which rational function has a graph that has vertical asymptotes at x 5a and

x5 2a, and a horizontal asymptote at y 50?

y 5(x2a)(_xx 1a) y 5 x2

x2 2a2

y 5 1

x2 2a2 y 5

x2a

x1a

Short Response

5. How many milliliters of 0.30% sugar solution must you add to 75 mL of 4% sugar solution to get a 0.50% sugar solution? Show your work.

A

H

C

G

75(0.04)10.003x

751 x 50.005

310.003x50.3751 0.005x x51312.5 mL [2] correct answer with work shown

(28)

8-3

Enrichment

_{Rational Functions and Their Graphs}

Other Asymptotes

Recall that a rational function does not have a horizontal asymptote if the degree of the numerator is greater than the degree of the denominator. If, however, the degree of the numerator is exactly one more than the degree of the denominator, then the graph of the function has a slant asymptote.

You can use long division to fi nd the equation of a slant asymptote. Th e equation of the slant asymptote is given by the quotient, disregarding the remainder.

1. Use long division to fi nd the slant asymptote of f(x) 5x_x2₁2₁x. Th e slant asymptote of the function is y5 . 2. Graph the slant asymptote on a coordinate grid.

3. Graph the vertical asymptote for the equation in Exercise 1 on the grid. 4. Copy and complete the table, and plot the points accordingly.

Connect with a smooth curve, being sure to draw near to all asymptotes.

5. Use long division to fi nd the slant asymptote of f(x) 5 x3

x2 11. Th e slant asymptote of the function is y5 .

6. Graph the slant asymptote on a new coordinate grid.

7. Find any vertical asymptotes for the equation in Exercise 5 and graph on the grid. 8. Copy and complete the table, and plot the points accordingly.

Connect with a smooth curve, being sure to draw near to all asymptotes.

9. Th e technique used to fi nd slant (linear) asymptotes works for rational functions in which the degree of the numerator is one more than the degree of the denominator. Use long

division to fi nd the non-linear asymptote of the rational function given by f(x)5x3_x11. Th e asymptote of the function is y= . Can you guess the

shape of this asymptote? x

f(x)

0 2

22

23 3

x

f(x)

0 2

22

23 3

x22

x

x2

4

⫺4

⫺4 4

x y

O

2

⫺2 ⫺2

2

x y

2₆ 26 0 2₃ 3₂

0 8₅

8 5

227 10

27 10

2

(29)

8-3

Reteaching

_{Rational Functions and Their Graphs}

A rational function may have one or more types of discontinuities: holes (removable points of discontinuity), vertical asymptotes (non-removable points of discontinuity), or a horizontal asymptote.

If Then Example

a is a zero with multiplicity

m in the numerator and multiplicity n in the de nominator, and m

$

n

hole at x5a

f(x)5(x25)(x16) (x 25) hole at x5 5

a is a zero of the denominator only, or a

is a zero with multiplicity

m in the numerator and multi plicity n in the denominator, and m

,

n

vertical asymptote

at x5a f(x)5

x2 x 23

vertical asymptote at x5 3

Let p 5degree o f numerator. Let q 5degree of denominator.

• m, n horizontal asymptote at y 50

f(x)5 4x2 7x2 12

horizontal asymptote at y5 4₇ • m. n no horizontal asymptote exists

• m5 n _{horizontal asymptote at}_y ₅a

b,

where a and b are coeffi cients of highest degree terms in numerator and denominator

Problem

What are the points of discontinuity of y 5x2 1x26

3x2 212 , if any?

Step 1 Factor the numerator and denominator completely. y 5 (x 22)(x 13) 3(x 22)(x12) Step 2 Look for values that are zeros of both the numerator and the denominator.

Th e function has a hole at x 52.

Step 3 Look for values that are zeros of the denominator only. Th e function has a vertical asymptote at x 5 22.

Step 4 Compare the degrees of the numerator and denominator. Th ey have the same degree. Th e function has a horizontal asymptote at y 51₃.

Exercises

Find the vertical asymptotes, holes, and horizontal asymptote for the graph of each rational function.

1. y5 x

x229 2. y 5

6x226

x 21 3. y5

4x 15 3x 12 horizontal asymptote:

y 5 0

vertical asymptotes: x 5 3, x 523; hole: x 5 1

(30)

8-3

Reteaching

(continued)

Rational Functions and Their Graphs

Before you try to sketch the graph of a rational function, get an idea of its general shape by identifying the graph’s holes, asymptotes, and intercepts.

Problem

What is the graph of the rational function y 5x_x1₁3₁?

Step 1 Identify any holes or asymptotes.

no holes; vertical asymptote at x5 21; horizontal asymptote at y 51₁5 1 Step 2 Identify any x- and y-intercepts.

x-intercepts occur when y 50.y-intercepts occur when x50.

x13

x11 50 y5

013 011

x13 50 y5 3

x5 23

x-intercept at 23 y-intercept at 3

Step 3 Sketch the asymptotes and Step 4 Make a table of values, plot the points,

intercepts. and sketch the graph.

Exercises

4. y5 4

x229 5. y5

x212x22

x21 O

⫺1

⫺3 ⫺2 2 3

2

1 3

⫺2 ⫺3

x

y _x _y

22

21.5

1

3

21

23

2

1.5 _O

⫺1

⫺3 ⫺2 2 3

2

1 3

⫺2 ⫺3

x y

⫺2 2 4

⫺4

⫺2

⫺3

1 2

x y

⫺4 4 6 8

⫺8 _⫺

4 4 8

(31)

8-4

Additional Vocabulary Support

_{Rational Expressions}

Th ere are two sets of cards that show how to simplify x224 x2 22x1 1 ?

x2 1 2x23 2x223x22. Th e set on the left explains the thinking. Th e set on the right shows the steps. Write the steps in the correct order.

Think Cards Write Cards

Factor the numerators and

denominators. (x

22)(x12)

(x 21)(x21) ?

(x 13)(x21)

(2x11)(x22)

(x 12)(x13)

(x21)(2x 11)

x2 ₂₄

x2 22x11?

x2 ₁_2x₂₃

2x2 23x22

(x22)(x12)

(x 21)(x21) ?

(x13)(x21)

(2x 11)(x22) Write the problem.

Write the remaining factors.

Divide out common factors.

Step 1

Step 2

Step 3

Step 4 First,

Second,

Third,

Fourth,

Think Write

write the problem.

factor the numerators and denominators.

divide out common factors.

write the remaining factors.

x2₂₄ x2₂_2x₁₁?

x2 ₁_2x₂₃ 2x2₂_3x₂₂

(x22)(x12) (x21)(x21)?

(x13)(x21) (2x11)(x22)

(x22)(x12) (x21)(x21) ?

(x13)(x21) (2x11)(x22)